ch11 - IE310 Operations Research Chapter 11 Integer...

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IE310. Operations Research Chapter 11. Integer Programming Sewoong Oh ISE department Lecture notes courtesy of Douglas M. King IE310 1 (C) Copyright 2012
IE310 2 INTEGER PROGRAMMING Recall that one of the requirements for a linear program was divisibility But what if we have a linear program requiring variables that are restricted to integer values? They are not linear programs Instead, we call them integer linear programs (or simply integer programs ) Simplex method on its own will not solve them Examples: Products can only be produced in integer numbers Number of employees hired must be integer (C) Copyright 2012 Divisibility: Each decision variable can take any numerical value (even non-integer values), as long as the decision variables satisfy the constraints.
IE310 3 INTEGER PROGRAMMING TERMINOLOGY Types of integer programs Binary integer program (BIP): Decision variables are all binary (0 or 1) Integer program (IP): Decision variables are all integers Mixed integer program (MIP): Some decision variables are integer (others not) To solve an IP, we will sometimes consider its LP relaxation , which discards the integer variable restrictions (i.e., same objective, constraints, and variables, but variables can take any real value) (C) Copyright 2012 Integer program: max 2 x 1 + x 2 s. t. 3 x 1 + 2 x 2 5 x 1 , x 2 0 x 1 , x 2 integer LP relaxation: max 2 x 1 + x 2 s. t. 3 x 1 + 2 x 2 5 x 1 , x 2 0 x 1 , x 2 integer
IE310 4 COMPARING THE IP AND ITS LP RELAXATION Optimal solution of IP: x 1 = 1 x 2 = 1 Objective = 3 Optimal solution of LP relaxation: x 1 = 5/3 x 2 = 0 Objective = 10/3 = 3.3333333 QUESTION: Why does the LP relaxation get a better objective value? (C) Copyright 2012 Integer program: max 2 x 1 + x 2 s. t. 3 x 1 + 2 x 2 5 x 1 , x 2 0 x 1 , x 2 integer LP relaxation: max 2 x 1 + x 2 s. t. 3 x 1 + 2 x 2 5 x 1 , x 2 0 x 1 , x 2 integer
IE310 5 x 1 x 2 Shaded region is feasible for the LP relaxation (C) Copyright 2012 Optimal solution Objective iso-line for Z = 3.33333
IE310 6 x 1 x 2 Integer points in feasible region are feasible in IP (C) Copyright 2012 This solution is no longer feasible! Objective iso-line for Z = 3.33333
IE310 7 x 1 x 2 Integer points in feasible region are feasible in IP (C) Copyright 2012 Optimal solution Objective iso-line for Z = 3
IE310 8 IP WITH SOLUTION ON THE INTERIOR In this example, the optimal solution for the IP remains on the boundary of the feasible region of the LP relaxation This will not be the case for all IPs Consider the following IP and its LP relaxation (changing the constraint RHS) (C) Copyright 2012 Integer program: max 2 x 1 + x 2 s. t. 3 x 1 + 2 x 2 5.5 x 1 , x 2 0 x 1 , x 2 integer LP relaxation: max 2 x 1 + x 2 s. t. 3 x 1 + 2 x 2 5.5 x 1 , x 2 0 x 1 , x 2 integer

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